The Néron Desingularization presented in the first section is a kind of
smoothification with respect to the inclusion \(B\rightarrow A^{\prime }\) . Now we want
to find a smoothification with respect to an \(A\) -morphism \(v:B\rightarrow A^{\prime }\)  in the case when \(A=k[x]_{(x)}\) , \(A^{\prime }=k[[ x ]]\) , \(k\)  being a field and \(x\)  a variable. Since we cannot give to a computer the whole informations concerning the coefficients of the formal power series \(v(Y_j)\) , it is better to work from the beginning with \(A\) -morphisms
\(v:B\rightarrow A^{\prime }/x^mA^{\prime }\)  for some \(m\gg 0\) . This is done in [1]}. Next we recall in sketch this construction since we need it for the algorithm. In [2]}, [3]} such algorithms are presented and even implemented for the so-called the General Néron Desingularization but in our case the things are simpler.
